arXiv: Geometric Topology最新文献_第3页

Combinatorics Encoding Geometry: The Legacy of Bill Thurston in the Story of One Theorem 组合学编码几何:比尔·瑟斯顿在《一个定理的故事》中的遗产

arXiv: Geometric Topology Pub Date : 2020-08-27 DOI: 10.1007/978-3-030-55928-1_5

Philip L. Bowers

引用次数: 10

Superbridge and bridge indices for knots 超桥和结的桥指数

arXiv: Geometric Topology Pub Date : 2020-08-14 DOI: 10.1142/S0218216521500097

C. Adams, Nikhil Agarwal, R. Allen, Tirasan Khandhawit, Alex Simons, Rebecca R. Winarski, Mary Wootters

引用次数: 3

Generalisations of Hecke algebras from Loop Braid Groups 环辫群中Hecke代数的推广

arXiv: Geometric Topology Pub Date : 2020-08-11 DOI: 10.2140/pjm.2023.323.31

C. Damiani, Paul Martin, E. Rowell

{"title":"Generalisations of Hecke algebras from Loop Braid Groups","authors":"C. Damiani, Paul Martin, E. Rowell","doi":"10.2140/pjm.2023.323.31","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.31","url":null,"abstract":"We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we introduce a class of local representations of the braid group derived from a meld of the Burau representation and the Rittenberg representations, here thus called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t in k$ the loop-Hecke parameter. We prove the following: \u0000$LH_n$ is finite dimensional over a field. \u0000The natural inclusion $LB_n rightarrow LB_{n+1}$ passes to an inclusion $SP_n rightarrow SP_{n+1}$. \u0000Over $k=mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. \u0000We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. \u0000The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. item For $t^2 neq 1$ then $LH_n cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). \u0000Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"2 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76036149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares 霍奇结构的变化和用三角形和正方形列举表面的平铺

arXiv: Geometric Topology Pub Date : 2020-07-08 DOI: 10.5802/JEP.159

Vincent Koziarz, Duc-Manh Nguyen

{"title":"Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares","authors":"Vincent Koziarz, Duc-Manh Nguyen","doi":"10.5802/JEP.159","DOIUrl":"https://doi.org/10.5802/JEP.159","url":null,"abstract":"Let $S$ be a connected closed oriented surface of genus $g$. Given a triangulation (resp. quadrangulation) of $S$, define the index of each of its vertices to be the number of edges originating from this vertex minus $6$ (resp. minus $4$). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If $kappa$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $min mathbb{Z}_{>0}$, denote by $mathscr{T}(kappa,m)$ (resp. $mathscr{Q}(kappa,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $kappa$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $kappa$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $kappa$ is divisible by $6$ (resp. by $4$), then $mathscr{T}(kappa,m)sim c_3(kappa)m^{2g+|kappa|-2}$ (resp. $mathscr{Q}(kappa,m) sim c_4(kappa)m^{2g+|kappa|-2}$), where $c_3(kappa) in mathbb{Q}cdot(sqrt{3}pi)^{2g+|kappa|-2}$ and $c_4(kappa)in mathbb{Q}cdotpi^{2g+|kappa|-2}$. The key ingredient of the proof is a result of J. Kollar on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of $pi$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"169 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75651003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 7

Constrained knots in lens spaces 透镜空间中的约束结

arXiv: Geometric Topology Pub Date : 2020-07-08 DOI: 10.2140/agt.2023.23.1097

Fan Ye

引用次数: 5

Palettes of Dehn colorings for spatial graphs and the classification of vertex conditions 空间图的Dehn着色调色板和顶点条件的分类

arXiv: Geometric Topology Pub Date : 2020-07-02 DOI: 10.1142/S0218216521500152

Kanako Oshiro, Natsumi Oyamaguchi

引用次数: 1

Cyclic branched covers of alternating knots 交替结的环状分枝盖

arXiv: Geometric Topology Pub Date : 2020-06-23 DOI: 10.5802/AHL.89

L. Paoluzzi